LITERATURE REVIEW
2.2. LITERATURE REVIEW
2.2.4. Updating of FE model
12
A cable-stayed bridge over the river Oglio in Italy (Benedettini and Gentile
2011)
EFDD, SSI
13
A stress-ribbon footbridge situated in the Faculty of Engineering of University of Porto (Hu et al. 2012)
SSI
14 Shanghai World Financial Center (Shi
et al. 2012) PP, RDT, Hilbert-Huang transform 15
Tamar Bridge - a long-span suspension in Southwest England (Cross et al.
2013)
SSI
16
The San Michele Bridge and A13 highway overpass in Bologna (Ubertini
et al. 2013)
FDD, SSI
Although many case studies are observed in the literature for various existing bridges as well as building structures, no large and complex bridge structure like the Saraighat Bridge is found to be investigated for modal parameters. Various strategies are found in literature for efficient modal identification against various unfavourable conditions, which are usually involved in OMA exercises. There are (a) implementation of multiple techniques, usually, both in time and frequency domains (b) identification of modal parameters in statistical way (central tendency, dispersion, confidence interval). Combined effort based on these major approaches is likely to have better acceptance of the identified modal parameters. Further, detailed study is not reported in the literature to find out the suitable size of Hankel Matrix, which is an important factor for efficient modal identification.
numerical FE model to an improved numerical model using the measured response of the corresponding real structural-system. Thus the improved numerical model becomes capable of simulating the corresponding real structural-system more accurately. The existing updating techniques can be broadly classified into three classes as: (i) direct updating techniques (ii) iterative techniques and (iii) frequency response based techniques. Present work deals with the direct model updating. Hence, literature review is kept here confined within the scope of direct updating only. FE model updating is quite a vast domain. However, an extensive review of literature on various types of updating techniques are presented by Mottershead and Friswell (1993); Friswell and Mottershead (1995); Ewins (2000b) etc.
An important as well as very popular branch of direct updating techniques is Lagrange multiplier based technique for direct updating. Optimization is carried out with the objective of minimal changes in the system matrices subjected to constraints, namely, modal orthogonality conditions in terms of mass and stiffness matrices; maintaining the symmetry of system matrices. All these Lagrange multiplier based techniques consider three quantities:
measured modal data (natural frequencies and mode shapes), analytical mass matrix and analytical stiffness matrix. Baruch and Bar-Itzhack (1978); Baruch (1978) performed direct updating using Lagrange multiplier considering exact mass matrix. Baruch (1982, 1984) described these methods as reference basis methods because one of the quantities out of measured modal data, analytical mass matrix and analytical stiffness matrix is assumed to be exact or the reference, while the remaining two quantities are updated. The measured eigenvectors are primarily corrected so that they become orthogonal with respected to the analytical mass matrix and the updated stiffness matrix. Therefore, the experimental mode shapes are not reproduced exactly. In a similar framework, as adopted by Baruch (1984), Berman and Nagy (1983) used the measured data as the reference and both the analytical mass and stiffness matrices in separate attempts. The measured modal data is exactly
reproduced by the updated matrices. The Berman and Nagy technique is presented with additional details in Section 6.2.1. Caesar (1986) further suggested a range of methods in a similar framework based on this Lagrange multiplier method where analytical mass and stiffness matrices are updated in separate attempts. Wei (1989; 1990a; 1990b) proposed an alternative way of updating where both the mass and stiffness matrices updated simultaneously using the measured eigenvector matrix as the reference. Constraints are considered as mass orthogonality and symmetry of the updated matrices. Usually the stiffness matrix elements are far larger than the mass matrix elements, hence this technique is considered to be biased providing higher weight to stiffness matrix.
The direct updating techniques have limitations with respect to the updated mass and stiffness matrices because of lesser physical significance due to non-compliance of the connectivity of nodes. Moreover the updated matrices are fully populated while ideally the system matrices contain non-zero elements in band-format along the leading diagonal. An attempt was carried out to update the only non-zero elements of the stiffness matrix based on the Lagrange multiplier method (Kabe 1985). Kabe's approch is able to preserve the sparsity pattern of the original stiffness matrix, however, large volume of computation is required in this approach. In another attempt to maintain the connectivity, Smith and Beattie (1991) considered quasi-Newton methods for updating of the stiffness matrix preserving the structural connectivity. However, the complicacy of interpretation of results is not much reduced since the higher frequency modes, which are usually not measured, contribute most to the stiffness matrix. Based on the Kabe’s (1985) study on maintaining connectivity, further attempt in this direction were observed in works carried out by Kammer (1988); Halevi and Bucher (2003). Kammer's presented a reformulation of Kabe's method to achieve more numrical stability and to provide more flexibility in defining weights in objective function.
However, computational cost was not improved.
Another type of popular direct updating technique is mentioned here as matrix mixing technique. This technique has the ability for exact reproduction of the measured modal frequencies and measured mode shapes. It utilizes the property of exact reconstruction of the mass and stiffness matrices using all the modal frequencies and mass normalized mode shapes with all DOF. Measurements of responses are carried out generally at limited numbers of coordinates. Moreover the numbers of identified modes are quite fewer than the numbers of analytical DOF. Such incompleteness in estimation of modal parameters gives raises various problems in estimating of mass and stiffness matrices. To overcome such difficulty, Thoren (1972) limited the number of DOF to equal the number of modes to maintain the modal matrix as square. Further, Ross (1971) added arbitrary linear independent vectors to the modal matrix to make it square as well as invertible. Further, Luk (1987) applied pseudo inversion to deal with the rectangular modal matrix. Matrix mixing technique (Caesar 1987;
Link et al. 1987) is usually considered as a development over the methods of Ross (1971) and Thoren (1972). The matrix mixing technique is presented in chapter 6 with further details.
Another class of direct FE model updating techniques has been emerged from control theory is known as eigen-structure assignment method. This method updates the system matrices (usually stiffness and damping matrices) to assign the measured eigenvalues and eigenvectors (natural frequencies, damping ratios and mode shapes). If the eigenvalues alone are assigned then this method may be regarded as well known pole placement technique. In an early study, Srinathkumar (1978) presented some guidelines to determine numbers of eigenvalues and eigenvectors elements may be assigned based on the concept of controllability and observability. The updated stiffness and damping matrices may not be necessarily symmetric. Minas and Inman (1988; 1990) carried out an attempt based on optimization procedure to maintain these matrices as symmetric. Overcoming this symmetry issue, Zimmerman and Windengren (1990) produced symmetric updated stiffness and
damping matrices directly by solving a Riccati type matrix equation. Early discussions on maintaining symmetry in the eigen-structure assignment problem can be found in the work of Andry et al. (1983).
Apart from various classes of direct updating techniques mentioned above, there have been some other relevant developments as well. Bucher and Braun (1993) presented a theoretical development showing in details how the necessary mass and stiffness modifications can be computed using modal test data only, even when the measured data are incomplete. Baruch and Bar-Itzhack (1998) presented a technique to force the measured modes for satisfying the theoretical condition of weighted orthogonality in an optimal way with the objective of minimum differences among the measured and corrected modes. Such corrected shapes can be further utilized in appropriate direct updating exercises. Friswell et al. (1998) extended the reference basis methods to update the damping matrix. The mass matrix was considered as correct and damping along with stiffness matrices were updated simultaneously. In a similar attempt, Kuo et al. (2006) presented finite element model updating problem which incorporated the measured modal data into the analytical finite element model with updating the damping and stiffness matrices. Kenigsbuch and Halevi (1998) presented the reference basis approach of direct updating in a general framework in terms of solving constrained optimization problem. Datta et al. (2000, 2002) presented the partial eigenstructure assignment problem. It was demonstrated that with the appropriate choice of gain and input influence matrices, certain eigenpairs of a vibrating system may be assigned while the other eigenpairs remain unchanged. Carvalho et al. (2007) presented computationally convenient a new method for finite element matrix updating problem in an undamped model with the capability of preventing the appearance of spurious modes.
Jacquelin et al. (2012) presented a direct updating technique based on a probabilistic approach where the natural frequencies and the eigenvectors of the system are measured and
assumed to be uncertain. Mao and Dai (2012) presented a new direct method for the finite element model updating technique which can preserve both no spill-over and positive semi- definiteness of the mass and stiffness matrices. Yuan (2013) recently presented a direct updating technique based on updating the analytical stiffness matrix. Updated matrix is attempted to retain many desired matrix properties satisfying the dynamic equation, symmetry, positive semi-definiteness and physical connectivity.
Although various updating techniques are available, the direct model updating is considered for the sample bridge. The sole aim of updating in the present study being efficient design of controller, direct updating is considered for their advantages like exact matching of the updated frequencies and mode shapes with the corresponding experimental frequencies and mode shapes. Moreover, applications of direct updating for large structural systems are not much observed in literature. Therefore it is a matter of great interest as well as challenge to consider this large bridge structure for direct FE model updating.